On the number of peaks of the eigenfunctions of the linearized Gel’fand problem

We derive a second order estimate for the first $$m$$m eigenvalues and eigenfunctions of the linearized Gel’fand problem associated to solutions which blow-up at m points. This allows us to determine, in some suitable situations, some qualitative properties of the first $$m$$m eigenfunctions as the number of points of concentration or the multiplicity of the eigenvalue.